Find the (h+t)(x) which is h(x)+t(x), the given is h(x)=(x+7 / 2-x) and t(x)=(x-2 / x+3)?step by step.

Let's find \((h + t)(x)\), which is the sum of the functions \(h(x)\) and \(t(x)\). The given functions are:

\[
h(x) = \left(x + \frac{7}{2} - x\right)
\]
\[
t(x) = \left(\frac{x - 2}{x + 3}\right)
\]

### Step 1: Simplify \(h(x)\)
First, let's simplify the expression for \(h(x)\):

\[
h(x) = x + \frac{7}{2} - x
\]

Here, the \(x\) and \(-x\) terms cancel each other out, so:

\[
h(x) = \frac{7}{2}
\]

This shows that \(h(x)\) is a constant function.

### Step 2: Combine \(h(x)\) and \(t(x)\)
Now, let's add \(h(x)\) and \(t(x)\) together to find \((h + t)(x)\):

\[
(h + t)(x) = h(x) + t(x) = \frac{7}{2} + \frac{x - 2}{x + 3}
\]

### Step 3: Find a Common Denominator
To add these two fractions, we need to express them with a common denominator. The common denominator will be \(2(x + 3)\):

\[
(h + t)(x) = \frac{7(x + 3)}{2(x + 3)} + \frac{2(x - 2)}{2(x + 3)}
\]

### Step 4: Simplify the Numerator
Now, simplify the expression by distributing and combining like terms in the numerator:

\[
(h + t)(x) = \frac{7(x + 3) + 2(x - 2)}{2(x + 3)}
\]

Distribute the 7 and 2:

\[
(h + t)(x) = \frac{7x + 21 + 2x - 4}{2(x + 3)}
\]

Combine the like terms:

\[
(h + t)(x) = \frac{9x + 17}{2(x + 3)}
\]

### Final Answer
The function \((h + t)(x)\) is:

\[
(h + t)(x) = \frac{9x + 17}{2(x + 3)}
\]

Would you like any further clarification on these steps or additional details?

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Here are 8 related questions to explore further:

1. What is the domain of the function \((h + t)(x)\)?
2. How would you subtract \(t(x)\) from \(h(x)\)?
3. How would you find the roots of the function \((h + t)(x)\)?
4. Can you find the inverse function of \((h + t)(x)\)?
5. What are the vertical and horizontal asymptotes of \((h + t)(x)\)?
6. How does the function \((h + t)(x)\) behave as \(x\) approaches infinity?
7. How would you graph the function \((h + t)(x)\)?
8. How would you multiply \(h(x)\) and \(t(x)\) instead of adding them?

**Tip:** When simplifying algebraic expressions, always look for opportunities to cancel out terms before combining fractions.

Learn how to find (h+t)(x) = h(x) + t(x) step by step with detailed algebraic solutions. Understand the addition of functions h(x) and t(x), involving fractional expressions and algebraic simplification. Suitable for high school students.

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